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Counting and Cardinality K.CC
Know number names and the count sequence.
1. Count to 100 by ones and by tens.
2. Count forward beginning from a given number within the known sequence (instead of having to begin at 1).
3. Write numbers from 0 to 20. Represent a number of objects with a written numeral 0-20 (with 0 representing a
count of no objects).
Count to tell the number of objects.
4. Understand the relationship between numbers and quantities; connect counting to cardinality.
a. When counting objects, say the number names in the standard order, pairing each object with one and
only one number name and each number name with one and only one object.
b. Understand that the last number name said tells the number of objects counted. The number of objects is
the same regardless of their arrangement or the order in which they were counted.
c. Understand that each successive number name refers to a quantity that is one larger.
5. Count to answer “how many?” questions about as many as 20 things arranged in a line, a rectangular array, or a
circle, or as many as 10 things in a scattered configuration; given a number from 1–20, count out that many objects.
Compare numbers.
6. Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in
another group, e.g., by using matching and counting strategies.1
1
Include groups with up to ten objects.
7. Compare two numbers between 1 and 10 presented as written numerals.
Number and Operations in Base Ten K.NBT
Work with numbers 11–19 to gain foundations for place value.
1. Compose and decompose numbers from 11 to 19 into ten ones and some further ones, e.g., by using objects or
drawings, and record each composition or decomposition by a drawing or equation (e.g., 18 = 10 + 8); understand
that these numbers are composed of ten ones and one, two, three, four, five, six, seven, eight, or nine ones.
Number and Operations in Base Ten 1.NBT
Extend the counting sequence.
1. Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number
of objects with a written numeral.
Understand place value.
2. Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the
following as special cases:
a. 10 can be thought of as a bundle of ten ones — called a “ten.”
b. The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or
nine ones.
c. The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or
nine tens (and 0 ones).
3. Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of
comparisons with the symbols >, =, and <.
Use place value understanding and properties of operations to add and subtract.
4. Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and
a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations,
and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the
reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes
it is necessary to compose a ten.
5. Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the
reasoning used.
6. Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 (positive or zero differences),
using concrete models or drawings and strategies based on place value, properties of operations, and/or the
relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.
Number and Operations in Base Ten 2.NBT
Understand place value.
1. Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706
equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:
a. 100 can be thought of as a bundle of ten tens — called a “hundred.”
b. The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven,
eight, or nine hundreds (and 0 tens and 0 ones).
2. Count within 1000; skip-count by 5s, 10s, and 100s.
3. Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.
4. Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and <
symbols to record the results of comparisons.
Use place value understanding and properties of operations to add and subtract.
5. Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the
relationship between addition and subtraction.
6. Add up to four two-digit numbers using strategies based on place value and properties of operations.
7. Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties
of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method.
Understand that in adding or subtracting three digit numbers, one adds or subtracts hundreds and hundreds, tens and
tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.
8. Mentally add 10 or 100 to a given number 100–900, and mentally subtract 10 or 100 from a given number 100–
900.
9. Explain why addition and subtraction strategies work, using place value and the properties of operations.3
3
Explanations may be supported by drawings or objects.
Number and Operations in Base Ten 3.NBT
Use place value understanding and properties of operations to perform multi-digit arithmetic.4
4
A range of algorithms may be used.
1. Use place value understanding to round whole numbers to the nearest 10 or 100.
2. Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of
operations, and/or the relationship between addition and subtraction.
3. Multiply one-digit whole numbers by multiples of 10 in the range 10–90 (e.g., 9 × 80, 5 × 60) using strategies
based on place value and properties of operations.
Number and Operations in Base Ten2 4.NBT
2
Grade 4 expectations in this domain are limited to whole numbers less than or equal to 1,000,000.
Generalize place value understanding for multi-digit whole numbers.
1. Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the
place to its right. For example, recognize that 700 ÷ 70 = 10 by applying concepts of place value and division.
2. Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare
two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the
results of comparisons.
3. Use place value understanding to round multi-digit whole numbers to any place.
Use place value understanding and properties of operations to perform multi-digit arithmetic.
4. Fluently add and subtract multi-digit whole numbers using the standard algorithm.
5. Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers,
using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using
equations, rectangular arrays, and/or area models.
6. Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using
strategies based on place value, the properties of operations, and/or the relationship between multiplication and
division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
Number and Operations in Base Ten 5.NBT
Understand the place value system.
1. Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the
place to its right and 1/10 of what it represents in the place to its left.
2. Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain
patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use wholenumber exponents to denote powers of 10.
3. Read, write, and compare decimals to thousandths.
a. Read and write decimals to thousandths using base-ten numerals, number names, and expanded form,
e.g., 347.392 = 3 × 100 + 4 × 10 + 7 × 1 + 3 × (1/10) + 9 × (1/100) + 2 × (1/1000).
b. Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and <
symbols to record the results of comparisons.
4. Use place value understanding to round decimals to any place.
Perform operations with multi-digit whole numbers and with decimals to hundredths.
5. Fluently multiply multi-digit whole numbers using the standard algorithm.
6. Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using
strategies based on place value, the properties of operations, and/or the relationship between multiplication and
division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
7. Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies
based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the
strategy to a written method and explain the reasoning used.
Number and Operations—Fractions5 3.NF
5
Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.
Develop understanding of fractions as numbers.
1. Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts;
understand a fraction a/b as the quantity formed by a parts of size 1/b.
2. Understand a fraction as a number on the number line; represent fractions on a number line diagram.
a. Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and
partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part
based at 0 locates the number 1/b on the number line.
b. Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that
the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.
3. Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.
a. Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number
line.
b. Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3). Explain why the fractions
are equivalent, e.g., by using a visual fraction model.
c. Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers.
Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a
number line diagram.
d. Compare two fractions with the same numerator or the same denominator by reasoning about their size.
Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the
results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual
fraction model.
Number and Operations—Fractions3 4.NF
3
Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.
Extend understanding of fraction equivalence and ordering.
1. Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with
attention to how the number and size of the parts differ even though the two fractions themselves are the same size.
Use this principle to recognize and generate equivalent fractions.
2. Compare two fractions with different numerators and different denominators, e.g., by creating common
denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are
valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or
<, and justify the conclusions, e.g., by using a visual fraction model.
Build fractions from unit fractions by applying and extending previous understandings of
operations on whole numbers.
3. Understand a fraction a/b with a > 1 as a sum of fractions 1/b.
a. Understand addition and subtraction of fractions as joining and separating parts referring to the same
whole.
b. Decompose a fraction into a sum of fractions with the same denominator in more than one way,
recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction
model. Examples: 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8.
c. Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an
equivalent fraction, and/or by using properties of operations and the relationship between addition and
subtraction.
d. Solve word problems involving addition and subtraction of fractions referring to the same whole and
having like denominators, e.g., by using visual fraction models and equations to represent the problem.
4. Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.
a. Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4
as the product 5 × (1/4), recording the conclusion by the equation 5/4 = 5 × (1/4).
b. Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a
whole number. For example, use a visual fraction model to express 3 × (2/5) as 6 × (1/5), recognizing this
product as 6/5. (In general, n × (a/b) = (n × a)/b.)
c. Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual
fraction models and equations to represent the problem. For example, if each person at a party will eat 3/8
of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be
needed? Between what two whole numbers does your answer lie?
Understand decimal notation for fractions, and compare decimal fractions.
5. Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to
add two fractions with respective denominators 10 and 100. 4 For example, express 3/10 as 30/100, and add
3/10 + 4/100 = 34/100.
4
Students who can generate equivalent fractions can develop strategies for adding fractions with unlike
denominators in general. But addition and subtraction with unlike denominators in general is not a
requirement at this grade.
6. Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100; describe a
length as 0.62 meters; locate 0.62 on a number line diagram.
7. Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only
when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and
justify the conclusions, e.g., by using a visual model.
Number and Operations—Fractions 5.NF
Use equivalent fractions as a strategy to add and subtract fractions.
1. Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with
equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators.
For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.)
2. Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases
of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark
fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. For example,
recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2.
Apply and extend previous understandings of multiplication and division to multiply and divide
fractions.
3. Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving
division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual
fraction models or equations to represent the problem. For example, interpret 3/4 as the result of dividing 3 by 4,
noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person
has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of
rice should each person get? Between what two whole numbers does your answer lie?
4. Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
a. Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of
a sequence of operations a × q ÷ b. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and
create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) =
ac/bd.)
b. Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate
unit fraction side lengths, and show that the area is the same as would be found by multiplying the side
lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as
rectangular areas.
5. Interpret multiplication as scaling (resizing), by:
a. Comparing the size of a product to the size of one factor on the basis of the size of the other factor,
without performing the indicated multiplication.
b. Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than
the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case);
explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the
given number; and relating the principle of fraction equivalence a/b = (n×a)/(n×b) to the effect of
multiplying a/b by 1.
6. Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction
models or equations to represent the problem.
7. Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole
numbers by unit fractions.1
1
Students able to multiply fractions in general can develop strategies to divide fractions in general, by
reasoning about the relationship between multiplication and division. But division of a fraction by a
fraction is not a requirement at this grade.
a. Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For
example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient.
Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because
(1/12) × 4 = 1/3.
b. Interpret division of a whole number by a unit fraction, and compute such quotients. For example,
create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the
relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 × (1/5) = 4.
c. Solve real world problems involving division of unit fractions by non-zero whole numbers and division
of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the
problem. For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate
equally? How many 1/3-cup servings are in 2 cups of raisins?
Ratios and Proportional Relationships 6.RP
Understand ratio concepts and use ratio reasoning to solve problems.
1. Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.
For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was
1 beak.” “For every vote candidate A received, candidate C received nearly three votes.”
2. Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the
context of a ratio relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is
3/4 cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.” 1
1
Expectations for unit rates in this grade are limited to non-complex fractions.
3. Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of
equivalent ratios, tape diagrams, double number line diagrams, or equations.
a. Make tables of equivalent ratios relating quantities with whole-number measurements, find missing
values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.
b. Solve unit rate problems including those involving unit pricing and constant speed. For example, if it
took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate
were lawns being mowed?
c. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity);
solve problems involving finding the whole, given a part and the percent.
d. Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when
multiplying or dividing quantities.
Ratios and Proportional Relationships 7.RP
Analyze proportional relationships and use them to solve real-world and mathematical problems.
1. Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities
measured in like or different units. For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate
as the complex fraction 1/2/1/4 miles per hour, equivalently 2 miles per hour.
2. Recognize and represent proportional relationships between quantities.
a. Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a
table or graphing on a coordinate plane and observing whether the graph is a straight line through the
origin.
b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal
descriptions of proportional relationships.
c. Represent proportional relationships by equations. For example, if total cost t is proportional to the
number n of items purchased at a constant price p, the relationship between the total cost and the number
of items can be expressed as t = pn.
d. Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation,
with special attention to the points (0, 0) and (1, r) where r is the unit rate.
3. Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax,
markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.
The Number System 6.NS
Apply and extend previous understandings of multiplication and division to divide fractions by
fractions.
1. Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions,
e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for
(2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and
division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much
chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3
of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?
Compute fluently with multi-digit numbers and find common factors and multiples.
2. Fluently divide multi-digit numbers using the standard algorithm.
3. Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.
4. Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple
of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers
1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example,
express 36 + 8 as 4 (9 + 2).
Apply and extend previous understandings of numbers to the system of rational numbers.
5. Understand that positive and negative numbers are used together to describe quantities having opposite directions
or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative
electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the
meaning of 0 in each situation.
6. Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes
familiar from previous grades to represent points on the line and in the plane with negative number coordinates.
a. Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line;
recognize that the opposite of the opposite of a number is the number itself, e.g., –(–3) = 3, and that 0 is its
own opposite.
b. Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate
plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by
reflections across one or both axes.
c. Find and position integers and other rational numbers on a horizontal or vertical number line diagram;
find and position pairs of integers and other rational numbers on a coordinate plane.
7. Understand ordering and absolute value of rational numbers.
a. Interpret statements of inequality as statements about the relative position of two numbers on a number
line diagram. For example, interpret –3 > –7 as a statement that –3 is located to the right of –7 on a
number line oriented from left to right.
b. Write, interpret, and explain statements of order for rational numbers in real-world contexts. For
example, write –3°C > –7°C to express the fact that –3°C is warmer than –7°C.
c. Understand the absolute value of a rational number as its distance from 0 on the number line; interpret
absolute value as magnitude for a positive or negative quantity in a real-world situation. For example, for
an account balance of –30 dollars, write |–30| = 30 to describe the size of the debt in dollars.
d. Distinguish comparisons of absolute value from statements about order. For example, recognize that an
account balance less than –30 dollars represents a debt greater than 30 dollars.
8. Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane.
Include use of coordinates and absolute value to find distances between points with the same first coordinate or the
same second coordinate.
The Number System 7.NS
Apply and extend previous understandings of operations with fractions to add, subtract, multiply,
and divide rational numbers.
1. Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers;
represent addition and subtraction on a horizontal or vertical number line diagram.
a. Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has
0 charge because its two constituents are oppositely charged.
b. Understand p + q as the number located a distance |q| from p, in the positive or negative direction
depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are
additive inverses.) Interpret sums of rational numbers by describing real-world contexts.
c. Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q). Show that
the distance between two rational numbers on the number line is the absolute value of their difference, and
apply this principle in real-world contexts.
d. Apply properties of operations as strategies to add and subtract rational numbers.
2. Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide
rational numbers.
a. Understand that multiplication is extended from fractions to rational numbers by requiring that
operations continue to satisfy the properties of operations, particularly the distributive property, leading to
products such as (–1)(–1) = 1 and the rules for multiplying signed numbers. Interpret products of rational
numbers by describing real-world contexts.
b. Understand that integers can be divided, provided that the divisor is not zero, and every quotient of
integers (with non-zero divisor) is a rational number. If p and q are integers, then –(p/q) = (–p)/q = p/(–q).
Interpret quotients of rational numbers by describing real world contexts.
c. Apply properties of operations as strategies to multiply and divide rational numbers.
d. Convert a rational number to a decimal using long division; know that the decimal form of a rational
number terminates in 0s or eventually repeats.
3. Solve real-world and mathematical problems involving the four operations with rational numbers. 1
1
Computations with rational numbers extend the rules for manipulating fractions to complex fractions.
The Number System 8.NS
Know that there are numbers that are not rational, and approximate them by rational numbers.
1. Know that numbers that are not rational are called irrational. Understand informally that every number has a
decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal
expansion which repeats eventually into a rational number.
2. Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them
approximately on a number line diagram, and estimate the value of expressions (e.g., √2). For example, by
truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how
to continue on to get better approximations.
High School – Number and Quantity
The Real Number System
N-RN
Extend the properties of exponents to rational exponents.
1.
Explain how the definition of the meaning of rational exponents follows from extending the properties of
integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For
example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5.
2.
Rewrite expressions involving radicals and rational exponents using the properties of exponents.
Use properties of rational and irrational numbers.
3.
Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an
irrational number is irrational; and that the product of a nonzero rational number and an irrational number is
irrational.
Quantities*
N-Q
Reason quantitatively and use units to solve problems.
1.
Use units as a way to understand problems and to guide the solution of multi-step problems; choose and
interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data
displays.
2.
Define appropriate quantities for the purpose of descriptive modeling.
3.
Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
The Complex Number System
N-CN
Perform arithmetic operations with complex numbers.
1.
Know there is a complex number i such that i 2 = -1, and every complex number has the form a + bi with a
and b real.
2.
Use the relation i 2 = -1 and the commutative, associative, and distributive properties to add, subtract, and
multiply complex numbers.
3.
(+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.
Represent complex numbers and their operations on the complex plane.
4.
(+) Represent complex numbers on the complex plane in rectangular and polar form (including real and
imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent
the same number.
5.
(+) Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the
complex plane; use properties of this representation for computation. For example, (-1 +3 i)3 = 8 because
(-1 +3 i) has modulus 2 and argument 120°.
6.
(+) Calculate the distance between numbers in the complex plane as the modulus of the difference, and the
midpoint of a segment as the average of the numbers at its endpoints.
Use complex numbers in polynomial identities and equations.
7.
Solve quadratic equations with real coefficients that have complex solutions.
8.
(+) Extend polynomial identities to the complex numbers. For example, rewrite x2 + 4 as (x + 2i)(x - 2i).
9.
(+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
Vector and Matrix Quantities
N-VM
Represent and model with vector quantities.
1.
(+) Recognize vector quantities as having both magnitude and direction. Represent vector quantities by
directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, |v|, ||v||, v).
2.
(+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a
terminal point.
3.
(+) Solve problems involving velocity and other quantities that can be represented by vectors.
Perform operations on vectors.
4.
(+) Add and subtract vectors.
a. Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude
of a sum of two vectors is typically not the sum of the magnitudes.
b. Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.
c. Understand vector subtraction v - w as v + (-w), where -w is the additive inverse of w, with the same
magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by
connecting the tips in the appropriate order, and perform vector subtraction component-wise.
5.
(+) Multiply a vector by a scalar.
a. Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction;
perform scalar multiplication component-wise, e.g., as c(vx, vy) = (cvx, cvy).
b. Compute the magnitude of a scalar multiple cv using ||cv|| = |c|v. Compute the direction of cv knowing that
when |c|v 0, the direction of cv is either along v (for c > 0) or against v (for c < 0).
Perform operations on matrices and use matrices in applications.
6.
(+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a
network.
7.
(+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are
doubled.
8.
(+) Add, subtract, and multiply matrices of appropriate dimensions.
9.
(+) Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a
commutative operation, but still satisfies the associative and distributive properties.
10. (+) Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the
role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a
multiplicative inverse.
11. (+) Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce
another vector. Work with matrices as transformations of vectors.
12. (+) Work with 2 × 2 matrices as transformations of the plane, and interpret the absolute value of the determinant
in terms of area.
Operations and Algebraic Thinking K.OA
Understand addition as putting together and adding to, and understand subtraction as taking apart
and taking from.
1. Represent addition and subtraction with objects, fingers, mental images, drawings 2, sounds (e.g., claps), acting out
situations, verbal explanations, expressions, or equations.
2
Drawings need not show details, but should show the mathematics in the problem.
(This applies wherever drawings are mentioned in the Standards.)
2. Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings
to represent the problem.
3. Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings,
and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).
4. For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using
objects or drawings, and record the answer with a drawing or equation.
5. Fluently add and subtract within 5.
Operations and Algebraic Thinking 1.OA
Represent and solve problems involving addition and subtraction.
1. Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from,
putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and
equations with a symbol for the unknown number to represent the problem. 2
2
See Glossary, Table 1.
2. Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by
using objects, drawings, and equations with a symbol for the unknown number to represent the problem.
Understand and apply properties of operations and the relationship between addition and
subtraction.
3. Apply properties of operations as strategies to add and subtract. 3
3
Students need not use formal terms for these properties.
Examples:
If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known. (Commutative property of addition.) To add 2 + 6 + 4, the
second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.)
4. Understand subtraction as an unknown-addend problem. For example, subtract 10 – 8 by finding the number that
makes 10 when added to 8.
Add and subtract within 20.
5. Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).
6. Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as
counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4
= 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12,
one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known
equivalent 6 + 6 + 1 = 12 + 1 = 13).
Work with addition and subtraction equations.
7. Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true
or false. For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 – 1,
5 + 2 = 2 + 5, 4 + 1 = 5 + 2.
8. Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. For
example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = ?– 3,
6 + 6 = �.
Operations and Algebraic Thinking 2.OA
Represent and solve problems involving addition and subtraction.
1. Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding
to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using
drawings and equations with a symbol for the unknown number to represent the problem.1
1
See Glossary, Table 1.
Add and subtract within 20.
2. Fluently add and subtract within 20 using mental strategies.2 By the end of Grade 2, know from memory all sums
of two one-digit numbers.
2
See standard 1.OA.6 for a list of mental strategies.
Work with equal groups of objects to gain foundations for multiplication.
3. Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects
or counting them by 2s; write an equation to express an even number as a sum of two equal addends.
4. Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5
columns; write an equation to express the total as a sum of equal addends.
Operations and Algebraic Thinking 3.OA
Represent and solve problems involving multiplication and division.
1. Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects
each. For example, describe a context in which a total number of objects can be expressed as 5 × 7.
2. Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share
when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into
equal shares of 8 objects each. For example, describe a context in which a number of shares or a number of groups
can be expressed as 56 ÷ 8.
3. Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays,
and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to
represent the problem.1
1
See Glossary, Table 2.
4. Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For
example, determine the unknown number that makes the equation true in each of the equations 8 × ? = 48,
5 = ? ÷ 3, 6 × 6 = ?.
Understand properties of multiplication and the relationship between multiplication and division.
5. Apply properties of operations as strategies to multiply and divide. 2
2
Students need not use formal terms for these properties.
Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.)
3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property
of multiplication.)
Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56.
(Distributive property.)
6. Understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding the number that makes
32 when multiplied by 8.
Multiply and divide within 100.
7. Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and
division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3,
know from memory all products of two one-digit numbers.
Solve problems involving the four operations, and identify and explain patterns in arithmetic.
8. Solve two-step word problems using the four operations. Represent these problems using equations with a letter
standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation
strategies including rounding.3
3
This standard is limited to problems posed with whole numbers and having whole number answers;
students should know how to perform operations in the conventional order when there are no parentheses to
specify a particular order (Order of Operations).
9. Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them
using properties of operations. For example, observe that 4 times a number is always even, and explain why 4 times
a number can be decomposed into two equal addends.
Operations and Algebraic Thinking 4.OA
Use the four operations with whole numbers to solve problems.
1. Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 × 7 as a statement that 35 is 5 times as
many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication
equations.
2. Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and
equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison
from additive comparison.1
1
See Glossary, Table 2.
3. Solve multistep word problems posed with whole numbers and having whole-number answers using the four
operations, including problems in which remainders must be interpreted. Represent these problems using equations
with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and
estimation strategies including rounding.
Gain familiarity with factors and multiples.
4. Find all factor pairs for a whole number in the range 1–100. Recognize that a whole number is a multiple of each
of its factors. Determine whether a given whole number in the range 1–100 is a multiple of a given one-digit
number. Determine whether a given whole number in the range 1–100 is prime or composite.
Generate and analyze patterns.
5. Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were
not explicit in the rule itself. For example, given the rule “Add 3” and the starting number 1, generate terms in the
resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain
informally why the numbers will continue to alternate in this way.
Operations and Algebraic Thinking 5.OA
Write and interpret numerical expressions.
1. Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.
2. Write simple expressions that record calculations with numbers, and interpret numerical expressions without
evaluating them. For example, express the calculation “add 8 and 7, then multiply by 2” as 2 × (8 + 7). Recognize
that 3 × (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or
product.
Analyze patterns and relationships.
3. Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding
terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a
coordinate plane. For example, given the rule “Add 3” and the starting number 0, and given the rule “Add 6” and
the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are
twice the corresponding terms in the other sequence. Explain informally why this is so.
Expressions and Equations 6.EE
Apply and extend previous understandings of arithmetic to algebraic expressions.
1. Write and evaluate numerical expressions involving whole-number exponents.
2. Write, read, and evaluate expressions in which letters stand for numbers.
a. Write expressions that record operations with numbers and with letters standing for numbers. For
example, express the calculation “Subtract y from 5” as 5 – y.
b. Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient,
coefficient); view one or more parts of an expression as a single entity. For example, describe the
expression 2 (8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two
terms.
c. Evaluate expressions at specific values of their variables. Include expressions that arise from formulas
used in real-world problems. Perform arithmetic operations, including those involving whole number
exponents, in the conventional order when there are no parentheses to specify a particular order (Order of
Operations). For example, use the formulas V = s3 and A = 6 s2 to find the volume and surface area of a
cube with sides of length s = 1/2.
3. Apply the properties of operations to generate equivalent expressions. For example, apply the distributive
property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to
the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y +
y to produce the equivalent expression 3y.
4. Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of
which value is substituted into them). For example, the expressions y + y + y and 3y are equivalent because they
name the same number regardless of which number y stands for.
Reason about and solve one-variable equations and inequalities.
5. Understand solving an equation or inequality as a process of answering a question: which values from a specified
set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a
specified set makes an equation or inequality true.
6. Use variables to represent numbers and write expressions when solving a real-world or mathematical problem;
understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a
specified set.
7. Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for
cases in which p, q and x are all nonnegative rational numbers.
8. Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical
problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of
such inequalities on number line diagrams.
Represent and analyze quantitative relationships between dependent and independent variables.
9. Use variables to represent two quantities in a real-world problem that change in relationship to one another; write
an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of
as the independent variable. Analyze the relationship between the dependent and independent variables using graphs
and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and
graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between
distance and time.
Expressions and Equations 7.EE
Use properties of operations to generate equivalent expressions.
1. Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational
coefficients.
2. Understand that rewriting an expression in different forms in a problem context can shed light on the problem and
how the quantities in it are related. For example, a + 0.05a = 1.05a means that “increase by 5%” is the same as
“multiply by 1.05.”
Solve real-life and mathematical problems using numerical and algebraic expressions and
equations.
3. Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any
form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate
with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using
mental computation and estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she
will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a
towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9
inches from each edge; this estimate can be used as a check on the exact computation.
4. Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and
inequalities to solve problems by reasoning about the quantities.
a. Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are
specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an
arithmetic solution, identifying the sequence of the operations used in each approach. For example, the
perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?
b. Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are
specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the
problem. For example: As a salesperson, you are paid $50 per week plus $3 per sale. This week you want
your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the
solutions.
Expressions and Equations 8.EE
Work with radicals and integer exponents.
1. Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example,
32 × 3–5 = 3–3 = 1/33 = 1/27.
2. Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is
a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes.
Know that √2 is irrational.
3. Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very
small quantities, and to express how many times as much one is than the other. For example, estimate the population
of the United States as 3 × 108 and the population of the world as 7 × 109, and determine that the world population
is more than 20 times larger.
4. Perform operations with numbers expressed in scientific notation, including problems where both decimal and
scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very
large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that
has been generated by technology.
Understand the connections between proportional relationships, lines, and linear equations.
5. Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different
proportional relationships represented in different ways. For example, compare a distance-time graph to a distancetime equation to determine which of two moving objects has greater speed.
6. Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line
in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line
intercepting the vertical axis at b.
Analyze and solve linear equations and pairs of simultaneous linear equations.
7. Solve linear equations in one variable.
a. Give examples of linear equations in one variable with one solution, infinitely many solutions, or no
solutions. Show which of these possibilities is the case by successively transforming the given equation
into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are
different numbers).
b. Solve linear equations with rational number coefficients, including equations whose solutions require
expanding expressions using the distributive property and collecting like terms.
8. Analyze and solve pairs of simultaneous linear equations.
a. Understand that solutions to a system of two linear equations in two variables correspond to points of
intersection of their graphs, because points of intersection satisfy both equations simultaneously.
b. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing
the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no
solution because 3x + 2y cannot simultaneously be 5and 6.
c. Solve real-world and mathematical problems leading to two linear equations in two variables. For
example, given coordinates for two pairs of points, determine whether the line through the first pair of
points intersects the line through the second pair.
High School – Algebra
Seeing Structure in Expressions
A-SSE
Interpret the structure of expressions
1.
Interpret expressions that represent a quantity in terms of its context.*
c. Interpret parts of an expression, such as terms, factors, and coefficients.
d. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example,
interpret P(1+r)n as the product of P and a factor not depending on P.
2.
Use the structure of an expression to identify ways to rewrite it. For example, see x4 - y4 as (x2)2 - (y2)2, thus
recognizing it as a difference of squares that can be factored as (x 2 - y2)(x2 + y2).
Write expressions in equivalent forms to solve problems
3.
Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity
represented by the expression.*
e. Factor a quadratic expression to reveal the zeros of the function it defines.
f. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it
defines.
g. Use the properties of exponents to transform expressions for exponential functions. For example the
expression 1.15 t can be rewritten as (1.151/12)12t  1.01212t to reveal the approximate equivalent monthly
interest rate if the annual rate is 15%.
4.
Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the
formula to solve problems. For example, calculate mortgage payments.*
Arithmetic with Polynomials and Rational Expressions
A-APR
Perform arithmetic operations on polynomials
1.
Understand that polynomials form a system analogous to the integers, namely, they are closed under the
operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
Understand the relationship between zeros and factors of polynomials
2.
Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division
by x - a is p(a), so p(a) = 0 if and only if (x - a) is a factor of p(x).
3.
Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a
rough graph of the function defined by the polynomial.
Use polynomial identities to solve problems
4.
Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial
identity (x2 + y2)2 = (x2 - y2)2 + (2xy)2 can be used to generate Pythagorean triples.
5.
(+) Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of x and y for a positive
integer n, where x and y are any numbers, with coefficients determined for example by Pascal’s Triangle. 1
The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument.
Rewrite rational expressions
6.
Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where
a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection,
long division, or, for the more complicated examples, a computer algebra system.
7.
(+) Understand that rational expressions form a system analogous to the rational numbers, closed under
addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply,
and divide rational expressions.
Creating Equations*
A-CED
Create equations that describe numbers or relationships
1.
Create equations and inequalities in one variable and use them to solve problems. Include equations arising
from linear and quadratic functions, and simple rational and exponential functions.
2.
Create equations in two or more variables to represent relationships between quantities; graph equations on
coordinate axes with labels and scales.
3.
Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret
solutions as viable or non- viable options in a modeling context. For example, represent inequalities describing
nutritional and cost constraints on combinations of different foods.
4.
Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For
example, rearrange Ohm's law V = IR to highlight resistance R.
Reasoning with Equations and Inequalities
A-REI
Understand solving equations as a process of reasoning and explain the reasoning
1.
Explain each step in solving a simple equation as following from the equality of numbers asserted at the
previous step, starting from the assumption that the original equation has a solution. Construct a viable
argument to justify a solution method.
2.
Solve simple rational and radical equations in one variable, and give examples showing how extraneous
solutions may arise.
Solve equations and inequalities in one variable
3.
Solve linear equations and inequalities in one variable, including equations with coefficients represented by
letters.
4.
Solve quadratic equations in one variable.
a. Use the method of completing the square to transform any quadratic equation in x into an equation of the
form (x - p)2 = q that has the same solutions. Derive the quadratic formula from this form.
b. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the
quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the
quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.
Solve systems of equations
5.
Prove that, given a system of two equations in two variables, replacing one equation by the sum of that
equation and a multiple of the other produces a system with the same solutions.
6.
Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear
equations in two variables.
7.
Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically
and graphically. For example, find the points of intersection between the line y = -3x and the circle x2 + y2 = 3.
8.
(+) Represent a system of linear equations as a single matrix equation in a vector variable.
9.
(+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for
matrices of dimension 3 × 3 or greater).
Represent and solve equations and inequalities graphically
10. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate
plane, often forming a curve (which could be a line).
11. Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are
the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the
functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are
linear, polynomial, rational, absolute value, exponential, and logarithmic functions.*
12. Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of
a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the
intersection of the corresponding half-planes.
Functions 8.F
Define, evaluate, and compare functions.
1. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set
of ordered pairs consisting of an input and the corresponding output.1
1
Function notation is not required in Grade 8.
2. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically
in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a
linear function represented by an algebraic expression, determine which function has the greater rate of change.
3. Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of
functions that are not linear. For example, the function A = s2 giving the area of a square as a function of its side
length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.
Use functions to model relationships between quantities.
4. Construct a function to model a linear relationship between two quantities. Determine the rate of change and
initial value of the function from a description of a relationship or from two (x, y) values, including reading these
from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the
situation it models, and in terms of its graph or a table of values.
5. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the
function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a
function that has been described verbally.
High School – Functions
Interpreting Functions
F-IF
Understand the concept of a function and use function notation
1.
Understand that a function from one set (called the domain) to another set (called the range) assigns to each
element of the domain exactly one element of the range. If f is a function and x is an element of its domain,
then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y =
f(x).
2.
Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function
notation in terms of a context.
3.
3. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the
integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1)
for n 1.
Interpret functions that arise in applications in terms of the context
4.
For a function that models a relationship between two quantities, interpret key features of graphs and tables in
terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative;
relative maximums and minimums; symmetries; end behavior; and periodicity.*
5.
Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.
For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory,
then the positive integers would be an appropriate domain for the function.*
6.
Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a
specified interval. Estimate the rate of change from a graph.*
Analyze functions using different representations
7.
Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and
using technology for more complicated cases.*
a. Graph linear and quadratic functions and show intercepts, maxima, and minima.
b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value
functions.
c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end
behavior.
d. (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available,
and showing end behavior.
e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric
functions, showing period, midline, and amplitude.
8.
Write a function defined by an expression in different but equivalent forms to reveal and explain different
properties of the function.
a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme
values, and symmetry of the graph, and interpret these in terms of a context.
b. Use the properties of exponents to interpret expressions for exponential functions. For example, identify
percent rate of change in functions such as y = (1.02) t, y = (0.97) t, y = (1.01)12t,
y = (1.2) t/10, and classify them as representing exponential growth or decay.
9.
Compare properties of two functions each represented in a different way (algebraically, graphically,
numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an
algebraic expression for another, say which has the larger maximum.
Building Functions
F-BF
Build a function that models a relationship between two quantities
1.
Write a function that describes a relationship between two quantities.*
a. Determine an explicit expression, a recursive process, or steps for calculation from a context.
b. Combine standard function types using arithmetic operations. For example, build a function that models
the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these
functions to the model.
c. (+) Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height,
and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the
location of the weather balloon as a function of time.
2.
Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model
situations, and translate between the two forms.*
Build new functions from existing functions
3.
Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k
(both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an
explanation of the effects on the graph using technology. Include recognizing even and odd functions from
their graphs and algebraic expressions for them.
4.
Find inverse functions.
a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for
the inverse. For example, f(x) =2 x3 or f(x) = (x+1)/(x-1) for x 1.
b. (+) Verify by composition that one function is the inverse of another.
c. (+) Read values of an inverse function from a graph or a table, given that the function has an inverse.
d. (+) Produce an invertible function from a non-invertible function by restricting the domain.
5.
(+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve
problems involving logarithms and exponents.
Linear, Quadratic, and Exponential Models*
F-LE
Construct and compare linear, quadratic, and exponential models and solve problems
1.
Distinguish between situations that can be modeled with linear functions and with exponential functions.
a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions
grow by equal factors over equal intervals.
b. Recognize situations in which one quantity changes aBuikt a constant rate per unit interval relative to
another.
c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval
relative to another.
2.
Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a
description of a relationship, or two input-output pairs (include reading these from a table).
3.
Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity
increasing linearly, quadratically, or (more generally) as a polynomial function.
4.
For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers and the
base b is 2, 10, or e; evaluate the logarithm using technology.
Interpret expressions for functions in terms of the situation they model
5.
Interpret the parameters in a linear or exponential function in terms of a context.
Trigonometric Functions
F-TF
Extend the domain of trigonometric functions using the unit circle
1.
Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
2.
Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real
numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.
3.
(+) Use special triangles to determine geometrically the values of sine, cosine, tangent for /3, /4 and /6, and
use the unit circle to express the values of sine, cosine, and tangent for -x, +x, and 2-x in terms of their
values for x, where x is any real number.
4.
(+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
Model periodic phenomena with trigonometric functions
5.
Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and
midline.*
6.
(+) Understand that restricting a trigonometric function to a domain on which it is always increasing or
always decreasing allows its inverse to be constructed.
7.
7. (+) Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the
solutions using technology, and interpret them in terms of the context.*
Prove and apply trigonometric identities
8.
Prove the Pythagorean identity sin 2() + cos2() = 1 and use it to find sin(), cos(), or tan() given sin(),
cos(), or tan() and the quadrant of the angle.
9.
(+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.
Measurement and Data K.MD
Describe and compare measurable attributes.
1. Describe measurable attributes of objects, such as length or weight. Describe several measurable attributes of a
single object.
2. Directly compare two objects with a measurable attribute in common, to see which object has “more of”/“less of”
the attribute, and describe the difference. For example, directly compare the heights of two children and describe
one child as taller/shorter.
Classify objects and count the number of objects in each category.
3. Classify objects into given categories; count the numbers of objects in each category and sort the categories by
count.3
3
Limit category counts to be less than or equal to 10.
Measurement and Data 1.MD
Measure lengths indirectly and by iterating length units.
1. Order three objects by length; compare the lengths of two objects indirectly by using a third object.
2. Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object
(the length unit) end to end; understand that the length measurement of an object is the number of same-size length
units that span it with no gaps or overlaps. Limit to contexts where the object being measured is spanned by a whole
number of length units with no gaps or overlaps.
Tell and write time.
3. Tell and write time in hours and half-hours using analog and digital clocks.
Represent and interpret data.
4. Organize, represent, and interpret data with up to three categories; ask and answer questions about the total
number of data points, how many in each category, and how many more or less are in one category than in another.
Measurement and Data 2.MD
Measure and estimate lengths in standard units.
1. Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks,
and measuring tapes.
2. Measure the length of an object twice, using length units of different lengths for the two measurements; describe
how the two measurements relate to the size of the unit chosen.
3. Estimate lengths using units of inches, feet, centimeters, and meters.
4. Measure to determine how much longer one object is than another, expressing the length difference in terms of a
standard length unit.
Relate addition and subtraction to length.
5. Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same
units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to
represent the problem.
6. Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding
to the numbers 0, 1, 2, ..., and represent whole-number sums and differences within 100 on a number line diagram.
Work with time and money.
7. Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.
8. Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols
appropriately. Example: If you have 2 dimes and 3 pennies, how many cents do you have?
Represent and interpret data.
9. Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making
repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal
scale is marked off in whole-number units.
10. Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories.
Solve simple put together, take-apart, and compare problems4 using information presented in a bar graph.
4
See Glossary, Table 1.
Measurement and Data 3.MD
Solve problems involving measurement and estimation of intervals of time, liquid volumes, and
masses of objects.
1. Tell and write time to the nearest minute and measure time intervals in minutes. Solve word problems involving
addition and subtraction of time intervals in minutes, e.g., by representing the problem on a number line diagram.
2. Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and
liters (l).6 Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are
given in the same units, e.g., by using drawings (such as a beaker with a measurement scale) to represent the
problem.7
6
Excludes compound units such as cm3 and finding the geometric volume of a container.
7
Excludes multiplicative comparison problems (problems involving notions of “times as much”; see Glossary,
Table 2).
Represent and interpret data.
3. Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and
two-step “how many more” and “how many less” problems using information presented in scaled bar graphs. For
example, draw a bar graph in which each square in the bar graph might represent 5 pets.
4. Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show
the data by making a line plot, where the horizontal scale is marked off in appropriate units— whole numbers,
halves, or quarters.
Geometric measurement: understand concepts of area and relate area to multiplication and to
addition.
5. Recognize area as an attribute of plane figures and understand concepts of area measurement.
a. A square with side length 1 unit, called “a unit square,” is said to have “one square unit” of area, and can
be used to measure area.
b. A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of
n square units.
6. Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units).
7. Relate area to the operations of multiplication and addition.
a. Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the
same as would be found by multiplying the side lengths.
b. Multiply side lengths to find areas of rectangles with whole number side lengths in the context of solving
real world and mathematical problems, and represent whole-number products as rectangular areas in
mathematical reasoning.
c. Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths
a and b + c is the sum of a × b and a × c. Use area models to represent the distributive property in
mathematical reasoning.
d. Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping
rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real world
problems.
Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish
between linear and area measures.
8. Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter
given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and
different areas or with the same area and different perimeters.
Measurement and Data 4.MD
Solve problems involving measurement and conversion of measurements from a larger unit to a
smaller unit.
1. Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr,
min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit.
Record measurement equivalents in a two column table. For example, know that 1 ft is 12 times as long as 1 in.
Express the length of a 4 ft snake as 48 in. Generate a conversion table for feet and inches listing the number pairs
(1, 12), (2, 24), (3, 36), ...
2. Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of
objects, and money, including problems involving simple fractions or decimals, and problems that require
expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using
diagrams such as number line diagrams that feature a measurement scale.
3. Apply the area and perimeter formulas for rectangles in real world and mathematical problems. For example, find
the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a
multiplication equation with an unknown factor.
Represent and interpret data.
4. Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Solve problems
involving addition and subtraction of fractions by using information presented in line plots. For example, from a line
plot find and interpret the difference in length between the longest and shortest specimens in an insect collection.
Geometric measurement: understand concepts of angle and measure angles.
5. Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and
understand concepts of angle measurement:
a. An angle is measured with reference to a circle with its center at the common endpoint of the rays, by
considering the fraction of the circular arc between the points where the two rays intersect the circle. An
angle that turns through 1/360 of a circle is called a “one-degree angle,” and can be used to measure angles.
b. An angle that turns through n one-degree angles is said to have an angle measure of n degrees.
6. Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure.
7. Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle
measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find
unknown angles on a diagram in real world and mathematical problems, e.g., by using an equation with a symbol for
the unknown angle measure.
Measurement and Data 5.MD
Convert like measurement units within a given measurement system.
1. Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5
cm to 0.05 m), and use these conversions in solving multi-step, real world problems.
Represent and interpret data.
2. Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on
fractions for this grade to solve problems involving information presented in line plots. For example, given different
measurements of liquid in identical beakers, find the amount of liquid each beaker would contain if the total amount
in all the beakers were redistributed equally.
Geometric measurement: understand concepts of volume and relate volume to multiplication and
to addition.
3. Recognize volume as an attribute of solid figures and understand concepts of volume measurement.
a. A cube with side length 1 unit, called a “unit cube,” is said to have “one cubic unit” of volume, and can
be used to measure volume.
b. A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume
of n cubic units.
4. Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.
5. Relate volume to the operations of multiplication and addition and solve real-world and mathematical problems
involving volume.
a. Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit
cubes, and show that the volume is the same as would be found by multiplying the edge lengths,
equivalently by multiplying the height by the area of the base. Represent threefold whole-number products
as volumes, e.g., to represent the associative property of multiplication.
b. Apply the formulas V = l × w × h and V = b × h for rectangular prisms to find volumes of right
rectangular prisms with whole number edge lengths in the context of solving real world and mathematical
problems.
c. Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right
rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve
real world problems.
Statistics and Probability 6.SP
Develop understanding of statistical variability.
1. Recognize a statistical question as one that anticipates variability in the data related to the question and accounts
for it in the answers. For example, “How old am I?” is not a statistical question, but “How old are the students in
my school?” is a statistical question because one anticipates variability in students’ ages.
2. Understand that a set of data collected to answer a statistical question has a distribution which can be described by
its center, spread, and overall shape.
3. Recognize that a measure of center for a numerical data set summarizes all of its values with a single number,
while a measure of variation describes how its values vary with a single number.
Summarize and describe distributions.
4. Display numerical data in plots on a number line, including dot plots, histograms, and box plots.
5. Summarize numerical data sets in relation to their context, such as by:
a. Reporting the number of observations.
b. Describing the nature of the attribute under investigation, including how it was measured and its units of
measurement.
c. Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or
mean absolute deviation), as well as describing any overall pattern and any striking deviations from the
overall pattern with reference to the context in which the data were gathered.
d. Relating the choice of measures of center and variability to the shape of the data distribution and the
context in which the data were gathered.
Statistics and Probability 7.SP
Use random sampling to draw inferences about a population.
1. Understand that statistics can be used to gain information about a population by examining a sample of the
population; generalizations about a population from a sample are valid only if the sample is representative of that
population. Understand that random sampling tends to produce representative samples and support valid inferences.
2. Use data from a random sample to draw inferences about a population with an unknown characteristic of interest.
Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions.
For example, estimate the mean word length in a book by randomly sampling words from the book; predict the
winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction
might be.
Draw informal comparative inferences about two populations.
3. Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities,
measuring the difference between the centers by expressing it as a multiple of a measure of variability. For example,
the mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer
team, about twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the
two distributions of heights is noticeable.
4. Use measures of center and measures of variability for numerical data from random samples to draw informal
comparative inferences about two populations. For example, decide whether the words in a chapter of a seventhgrade science book are generally longer than the words in a chapter of a fourth-grade science book.
Investigate chance processes and develop, use, and evaluate probability models.
5. Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the
event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a
probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely
event.
6. Approximate the probability of a chance event by collecting data on the chance process that produces it and
observing its long-run relative frequency, and predict the approximate relative frequency given the probability. For
example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but
probably not exactly 200 times.
7. Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to
observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.
a. Develop a uniform probability model by assigning equal probability to all outcomes, and use the model
to determine probabilities of events. For example, if a student is selected at random from a class, find the
probability that Jane will be selected and the probability that a girl will be selected.
b. Develop a probability model (which may not be uniform) by observing frequencies in data generated
from a chance process.
For example, find the approximate probability that a spinning penny will land heads up or that a tossed
paper cup will land open-end down. Do the outcomes for the spinning penny appear to be equally likely
based on the observed frequencies?
8. Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.
a. Understand that, just as with simple events, the probability of a compound event is the fraction of
outcomes in the sample space for which the compound event occurs.
b. Represent sample spaces for compound events using methods such as organized lists, tables and tree
diagrams. For an event described in everyday language (e.g., “rolling double sixes”), identify the outcomes
in the sample space which compose the event.
c. Design and use a simulation to generate frequencies for compound events. For example, use random
digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood,
what is the probability that it will take at least 4donors to find one with type A blood?
Statistics and Probability 8.SP
Investigate patterns of association in bivariate data.
1. Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between
two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and
nonlinear association.
2. Know that straight lines are widely used to model relationships between two quantitative variables. For scatter
plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging
the closeness of the data points to the line.
3. Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the
slope and intercept. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as
meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height.
4. Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies
and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two
categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to
describe possible association between the two variables. For example, collect data from students in your class on
whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there
evidence that those who have a curfew also tend to have chores?
High School – Probability and Statistics
Interpreting Categorical and Quantitative Data
S-ID
Summarize, represent, and interpret data on a single count or measurement variable
1.
Represent data with plots on the real number line (dot plots, histograms, and box plots).
2.
Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread
(interquartile range, standard deviation) of two or more different data sets.
3.
Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible
effects of extreme data points (outliers).
4.
Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population
percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators,
spreadsheets, and tables to estimate areas under the normal curve.
Summarize, represent, and interpret data on two categorical and quantitative variables
5.
Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in
the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible
associations and trends in the data.
6.
Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given
functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential
models.
b. Informally assess the fit of a function by plotting and analyzing residuals.
c. Fit a linear function for a scatter plot that suggests a linear association.
Interpret linear models
7.
Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the
data.
8.
Compute (using technology) and interpret the correlation coefficient of a linear fit.
9.
Distinguish between correlation and causation.
Making Inferences and Justifying Conclusions
S-IC
Understand and evaluate random processes underlying statistical experiments
1.
Understand statistics as a process for making inferences about population parameters based on a random
sample from that population.
2.
Decide if a specified model is consistent with results from a given data-generating process, e.g., using
simulation. For example, a model says a spinning coin falls heads up with probability 0.5. Would a result of 5
tails in a row cause you to question the model?
Make inferences and justify conclusions from sample surveys, experiments, and observational studies
3.
Recognize the purposes of and differences among sample surveys, experiments, and observational studies;
explain how randomization relates to each.
4.
Use data from a sample survey to estimate a population mean or proportion; develop a margin of error
through the use of simulation models for random sampling.
5.
Use data from a randomized experiment to compare two treatments; use simulations to decide if differences
between parameters are significant.
6.
Evaluate reports based on data.
Conditional Probability and the Rules of Probability
S-CP
Understand independence and conditional probability and use them to interpret data
1.
Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the
outcomes, or as unions, intersections, or complements of other events ("or," "and," "not").
2.
Understand that two events A and B are independent if the probability of A and B occurring together is the
product of their probabilities, and use this characterization to determine if they are independent.
3.
Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B
as saying that the conditional probability of A given B is the same as the probability of A, and the conditional
probability of B given A is the same as the probability of B.
4.
Construct and interpret two-way frequency tables of data when two categories are associated with each object
being classified. Use the two-way table as a sample space to decide if events are independent and to
approximate conditional probabilities. For example, collect data from a random sample of students in your
school on their favorite subject among math, science, and English. Estimate the probability that a randomly
selected student from your school will favor science given that the student is in tenth grade. Do the same for
other subjects and compare the results.
5.
Recognize and explain the concepts of conditional probability and independence in everyday language and
everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the
chance of being a smoker if you have lung cancer.
Use the rules of probability to compute probabilities of compound events in a uniform probability model
6.
Find the conditional probability of A given B as the fraction of B's outcomes that also belong to A, and interpret
the answer in terms of the model.
7.
Apply the Addition Rule, P(A or B) = P(A) + P(B) - P(A and B), and interpret the answer in terms of the
model.
8.
(+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) =
P(B)P(A|B), and interpret the answer in terms of the model.
9.
(+) Use permutations and combinations to compute probabilities of compound events and solve problems.
Using Probability to make decisions
S-MD
Calculate expected values and use them to solve problems
1.
(+) Define a random variable for a quantity of interest by assigning a numerical value to each event in a
sample space; graph the corresponding probability distribution using the same graphical displays as for data
distributions.
2.
(+) Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.
3.
(+) Develop a probability distribution for a random variable defined for a sample space in which theoretical
probabilities can be calculated; find the expected value. For example, find the theoretical probability
distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice
test where each question has four choices, and find the expected grade under various grading schemes.
4.
(+) Develop a probability distribution for a random variable defined for a sample space in which probabilities
are assigned empirically; find the expected value. For example, find a current data distribution on the number
of TV sets per household in the United States, and calculate the expected number of sets per household. How
many TV sets would you expect to find in 100 randomly selected households?
Use probability to evaluate outcomes of decisions
5.
(+) Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected
values.
a. Find the expected payoff for a game of chance. For example, find the expected winnings from a state
lottery ticket or a game at a fast- food restaurant.
b. Evaluate and compare strategies on the basis of expected values. For example, compare a high-deductible
versus a low-deductible automobile insurance policy using various, but reasonable, chances of having a
minor or a major accident.
6.
(+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).
7.
(+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a
hockey goalie at the end of a game).
Geometry K.G
Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones,
cylinders, and spheres).
1. Describe objects in the environment using names of shapes, and describe the relative positions of these objects
using terms such as above, below, beside, in front of, behind, and next to.
2. Correctly name shapes regardless of their orientations or overall size.
3. Identify shapes as two-dimensional (lying in a plane, “flat”) or three-dimensional (“solid”).
Analyze, compare, create, and compose shapes.
4. Analyze and compare two- and three-dimensional shapes, in different sizes and orientations, using informal
language to describe their similarities, differences, parts (e.g., number of sides and vertices/“corners”) and other
attributes (e.g., having sides of equal length).
5. Model shapes in the world by building shapes from components (e.g., sticks and clay balls) and drawing shapes.
6. Compose simple shapes to form larger shapes. For example, “Can you join these two triangles with full sides
touching to make a rectangle?”
Geometry 1.G
Reason with shapes and their attributes.
1. Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes
(e.g., color, orientation, overall size); build and draw shapes to possess defining attributes.
2. Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or
three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create
a composite shape, and compose new shapes from the composite shape.4
4
Students do not need to learn formal names such as “right rectangular prism.”
3. Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths,
and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the
shares. Understand for these examples that decomposing into more equal shares creates smaller shares.
Geometry 2.G
Reason with shapes and their attributes.
1. Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of
equal faces.5 Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.
5
Sizes are compared directly or visually, not compared by measuring.
2. Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.
3. Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves,
thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal
shares of identical wholes need not have the same shape.
Geometry 3.G
Reason with shapes and their attributes.
1. Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g.,
having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). Recognize
rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not
belong to any of these subcategories.
2. Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. For
example, partition a shape into 4 parts with equal area, and describe the area of each part as 1/4 of the area of the
shape.
Geometry 4.G
Draw and identify lines and angles, and classify shapes by properties of their lines and angles.
1. Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify
these in two-dimensional figures.
2. Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the
presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles.
3. Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be
folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry.
Geometry 5.G
Graph points on the coordinate plane to solve real-world and mathematical problems.
1. Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the
lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an
ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the
origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second
axis, with the convention that the names of the two axes and the coordinates correspond (e.g., x-axis and xcoordinate, y-axis and y-coordinate).
2. Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane,
and interpret coordinate values of points in the context of the situation.
Classify two-dimensional figures into categories based on their properties.
3. Understand that attributes belonging to a category of two dimensional figures also belong to all subcategories of
that category.
For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles.
4. Classify two-dimensional figures in a hierarchy based on properties.
Geometry 6.G
Solve real-world and mathematical problems involving area, surface area, and volume.
1. Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles
or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and
mathematical problems.
2. Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the
appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the
edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with
fractional edge lengths in the context of solving real-world and mathematical problems.
3. Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a
side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the
context of solving real-world and mathematical problems.
4. Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the
surface area of these figures. Apply these techniques in the context of solving real-world and mathematical
problems.
Geometry 7.G
Draw, construct, and describe geometrical figures and describe the relationships between them.
1. Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from
a scale drawing and reproducing a scale drawing at a different scale.
2. Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus
on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique
triangle, more than one triangle, or no triangle.
3. Describe the two-dimensional figures that result from slicing three dimensional figures, as in plane sections of
right rectangular prisms and right rectangular pyramids.
Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.
4. Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal
derivation of the relationship between the circumference and area of a circle.
5. Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and
solve simple equations for an unknown angle in a figure.
6. Solve real-world and mathematical problems involving area, volume and surface area of two- and threedimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.
Geometry 8.G
Understand congruence and similarity using physical models, transparencies, or geometry
software.
1. Verify experimentally the properties of rotations, reflections, and translations:
a. Lines are taken to lines, and line segments to line segments of the same length.
b. Angles are taken to angles of the same measure.
c. Parallel lines are taken to parallel lines.
2. Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a
sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits
the congruence between them.
3. Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using
coordinates.
4. Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a
sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a
sequence that exhibits the similarity between them.
5. Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles
created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For
example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and
give an argument in terms of transversals why this is so.
Understand and apply the Pythagorean Theorem.
6. Explain a proof of the Pythagorean Theorem and its converse.
7. Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and
mathematical problems in two and three dimensions.
8. Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.
9. Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and
mathematical problems.
High School – Geometry
Congruence
G-CO
Experiment with transformations in the plane
1.
Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the
undefined notions of point, line, distance along a line, and distance around a circular arc.
2.
Represent transformations in the plane using, e.g., transparencies and geometry software; describe
transformations as functions that take points in the plane as inputs and give other points as outputs. Compare
transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
3.
Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry
it onto itself.
4.
Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines,
parallel lines, and line segments.
5.
Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph
paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure
onto another.
Understand congruence in terms of rigid motions
6.
Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid
motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide
if they are congruent.
7.
Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and
only if corresponding pairs of sides and corresponding pairs of angles are congruent.
8.
Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of
congruence in terms of rigid motions.
Prove geometric theorems
9.
Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal
crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points
on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.
10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base
angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel
to the third side and half the length; the medians of a triangle meet at a point.
11. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are
congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms
with congruent diagonals.
Make geometric constructions
12. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string,
reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle;
bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular
bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.
13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
Similarity, Right Triangles, and Trigonometry
G-SRT
Understand similarity in terms of similarity transformations
1.
Verify experimentally the properties of dilations given by a center and a scale factor:
a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line
passing through the center unchanged.
b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
2.
Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are
similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all
corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
3.
Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
Prove theorems involving similarity
4.
Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other
two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
5.
Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric
figures.
Define trigonometric ratios and solve problems involving right triangles
6.
Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to
definitions of trigonometric ratios for acute angles.
7.
Explain and use the relationship between the sine and cosine of complementary angles.
8.
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.*
Apply trigonometry to general triangles
9.
(+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex
perpendicular to the opposite side.
10. (+) Prove the Laws of Sines and Cosines and use them to solve problems.
11. (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and
non-right triangles (e.g., surveying problems, resultant forces).
Circles
G-C
Understand and apply theorems about circles
1.
Prove that all circles are similar.
2.
Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between
central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a
circle is perpendicular to the tangent where the radius intersects the circle.
3.
Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral
inscribed in a circle.
4.
(+) Construct a tangent line from a point outside a given circle to the circle.
Find arc lengths and areas of sectors of circles
5.
Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius,
and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a
sector.
Expressing Geometric Properties with Equations
G-GPE
Translate between the geometric description and the equation for a conic section
1.
Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the
square to find the center and radius of a circle given by an equation.
2.
Derive the equation of a parabola given a focus and directrix.
3.
(+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of
distances from the foci is constant.
Use coordinates to prove simple geometric theorems algebraically
4.
Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a
figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point
(1,3) lies on the circle centered at the origin and containing the point (0, 2).
5.
Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find
the equation of a line parallel or perpendicular to a given line that passes through a given point).
6.
Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
7.
Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance
formula.*
Geometric measurement and dimension
G-GMD
Explain volume formulas and use them to solve problems
1.
Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a
cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments.
2.
(+) Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other
solid figures.
3.
Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.*
Visualize relationships between two-dimensional and three-dimensional objects
4.
Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify threedimensional objects generated by rotations of two-dimensional objects.
Modeling with Geometry
G-MG
Apply geometric concepts in modeling situations
1.
Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a
human torso as a cylinder).*
2.
Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile,
BTUs per cubic foot).*
3.
Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical
constraints or minimize cost; working with typographic grid systems based on ratios).*
High School—Modeling
Modeling links classroom mathematics and statistics to everyday life, work, and decision-making. Modeling is the
process of choosing and using appropriate mathematics and statistics to analyze empirical situations, to understand
them better, and to improve decisions. Quantities and their relationships in physical, economic, public policy, social,
and everyday situations can be modeled using mathematical and statistical methods. When making mathematical
models, technology is valuable for varying assumptions, exploring consequences, and comparing predictions with
data.
A model can be very simple, such as writing total cost as a product of unit price and number bought, or using a
geometric shape to describe a physical object like a coin. Even such simple models involve making choices. It is up
to us whether to model a coin as a three-dimensional cylinder, or whether a two-dimensional disk works well
enough for our purposes. Other situations—modeling a delivery route, a production schedule, or a comparison of
loan amortizations—need more elaborate models that use other tools from the mathematical sciences. Real-world
situations are not organized and labeled for analysis; formulating tractable models, representing such models, and
analyzing them is appropriately a creative process. Like every such process, this depends on acquired expertise as
well as creativity.
Some examples of such situations might include:
 Estimating how much water and food is needed for emergency relief in a devastated city of 3 million people,
and how it might be distributed.
 Planning a table tennis tournament for 7 players at a club with 4 tables, where each player plays against each
other player.
 Designing the layout of the stalls in a school fair so as to raise as much money as possible.
 Analyzing stopping distance for a car.
 Modeling savings account balance, bacterial colony growth, or investment growth.
 Engaging in critical path analysis, e.g., applied to turnaround of an aircraft at an airport.
 Analyzing risk in situations such as extreme sports, pandemics, and terrorism.
 Relating population statistics to individual predictions.
In situations like these, the models devised depend on a number of factors: How precise an answer do we want or
need? What aspects of the situation do we most need to understand, control, or optimize? What resources of time
and tools do we have? The range of models that we can create and analyze is also constrained by the limitations of
our mathematical, statistical, and technical skills, and our ability to recognize significant variables and relationships
among them. Diagrams of various kinds, spreadsheets and other technology, and algebra are powerful tools for
understanding and solving problems drawn from different types of real-world situations.
One of the insights provided by mathematical modeling is that essentially the same mathematical or statistical
structure can sometimes model seemingly different
situations. Models can also shed light on the
mathematical structures themselves, for example, as
when a model of bacterial growth makes more vivid
the explosive growth of the exponential function.
The basic modeling cycle is summarized in the
diagram. It involves (1) identifying variables in the situation and selecting those that represent essential features, (2)
formulating a model by creating and selecting geometric, graphical, tabular, algebraic, or statistical representations
that describe relationships between the variables, (3) analyzing and performing operations on these relationships to
draw conclusions, (4) interpreting the results of the mathematics in terms of the original situation, (5) validating the
conclusions by comparing them with the situation, and then either improving the model or, if it is acceptable, (6)
reporting on the conclusions and the reasoning behind them. Choices, assumptions, and approximations are present
throughout this cycle.
In descriptive modeling, a model simply describes the phenomena or summarizes them in a compact form. Graphs
of observations are a familiar descriptive model—for example, graphs of global temperature and atmospheric CO 2
over time.
Analytic modeling seeks to explain data on the basis of deeper theoretical ideas, albeit with parameters that are
empirically based; for example, exponential growth of bacterial colonies (until cut-off mechanisms such as pollution
or starvation intervene) follows from a constant reproduction rate. Functions are an important tool for analyzing
such problems.
Graphing utilities, spreadsheets, computer algebra systems, and dynamic geometry software are powerful tools that
can be used to model purely mathematical phenomena (e.g., the behavior of polynomials) as well as physical
phenomena.
Modeling Standards Modeling is best interpreted not as a collection of isolated topics but rather in relation to
other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling
standards appear throughout the high school standards indicated by a star symbol (*).